3.656 \(\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx\)

Optimal. Leaf size=52 \[ \frac {\sqrt {1-2 x^2} \sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {x-1} \sqrt {x+1} \sqrt {2 x^2-1}} \]

[Out]

EllipticF(x,2^(1/2))*(-2*x^2+1)^(1/2)*(-x^2+1)^(1/2)/(-1+x)^(1/2)/(1+x)^(1/2)/(2*x^2-1)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {519, 421, 419} \[ \frac {\sqrt {1-2 x^2} \sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {x-1} \sqrt {x+1} \sqrt {2 x^2-1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + x]*Sqrt[1 + x]*Sqrt[-1 + 2*x^2]),x]

[Out]

(Sqrt[1 - 2*x^2]*Sqrt[1 - x^2]*EllipticF[ArcSin[x], 2])/(Sqrt[-1 + x]*Sqrt[1 + x]*Sqrt[-1 + 2*x^2])

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 421

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d*x^2)/c]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx &=\frac {\sqrt {-1+x^2} \int \frac {1}{\sqrt {-1+x^2} \sqrt {-1+2 x^2}} \, dx}{\sqrt {-1+x} \sqrt {1+x}}\\ &=\frac {\left (\sqrt {1-2 x^2} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {1-2 x^2} \sqrt {-1+x^2}} \, dx}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}}\\ &=\frac {\left (\sqrt {1-2 x^2} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-2 x^2} \sqrt {1-x^2}} \, dx}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}}\\ &=\frac {\sqrt {1-2 x^2} \sqrt {1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}}\\ \end {align*}

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Mathematica [B]  time = 0.24, size = 107, normalized size = 2.06 \[ -\frac {2 (x-1)^{3/2} \sqrt {\frac {x+1}{1-x}} \sqrt {\frac {1-2 x^2}{(x-1)^2}} F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt {2}+2+\frac {1}{x-1}}}{2^{3/4}}\right )|4 \left (-4+3 \sqrt {2}\right )\right )}{\sqrt {3+2 \sqrt {2}} \sqrt {x+1} \sqrt {2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Sqrt[-1 + x]*Sqrt[1 + x]*Sqrt[-1 + 2*x^2]),x]

[Out]

(-2*(-1 + x)^(3/2)*Sqrt[(1 + x)/(1 - x)]*Sqrt[(1 - 2*x^2)/(-1 + x)^2]*EllipticF[ArcSin[Sqrt[2 + Sqrt[2] + (-1
+ x)^(-1)]/2^(3/4)], 4*(-4 + 3*Sqrt[2])])/(Sqrt[3 + 2*Sqrt[2]]*Sqrt[1 + x]*Sqrt[-1 + 2*x^2])

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fricas [F]  time = 1.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, x^{2} - 1} \sqrt {x + 1} \sqrt {x - 1}}{2 \, x^{4} - 3 \, x^{2} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/(1+x)^(1/2)/(2*x^2-1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(2*x^2 - 1)*sqrt(x + 1)*sqrt(x - 1)/(2*x^4 - 3*x^2 + 1), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{2} - 1} \sqrt {x + 1} \sqrt {x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/(1+x)^(1/2)/(2*x^2-1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(2*x^2 - 1)*sqrt(x + 1)*sqrt(x - 1)), x)

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maple [A]  time = 0.03, size = 58, normalized size = 1.12 \[ \frac {\sqrt {x -1}\, \sqrt {x +1}\, \sqrt {2 x^{2}-1}\, \sqrt {-x^{2}+1}\, \sqrt {-2 x^{2}+1}\, \EllipticF \left (x , \sqrt {2}\right )}{2 x^{4}-3 x^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x-1)^(1/2)/(x+1)^(1/2)/(2*x^2-1)^(1/2),x)

[Out]

(x-1)^(1/2)*(x+1)^(1/2)*(2*x^2-1)^(1/2)/(2*x^4-3*x^2+1)*(-x^2+1)^(1/2)*(-2*x^2+1)^(1/2)*EllipticF(x,2^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{2} - 1} \sqrt {x + 1} \sqrt {x - 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(1/2)/(1+x)^(1/2)/(2*x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(2*x^2 - 1)*sqrt(x + 1)*sqrt(x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {2\,x^2-1}\,\sqrt {x-1}\,\sqrt {x+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2*x^2 - 1)^(1/2)*(x - 1)^(1/2)*(x + 1)^(1/2)),x)

[Out]

int(1/((2*x^2 - 1)^(1/2)*(x - 1)^(1/2)*(x + 1)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {x + 1} \sqrt {2 x^{2} - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)**(1/2)/(1+x)**(1/2)/(2*x**2-1)**(1/2),x)

[Out]

Integral(1/(sqrt(x - 1)*sqrt(x + 1)*sqrt(2*x**2 - 1)), x)

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